ON TRANSVERSALITY WITH DEFICIENCY AND A CONJECTURE OF SARD

Let f : M --> N be a C-tau map between C-tau manifolds (tau greater th an or equal to 1) and K a C-tau manifold. In this paper, by using the Sard theorem, we study the topological properties of the space of C-ta u maps g : K --> N which satisfy a certain transversality condition wi th respect to f in a weak sense. As an application, by considering the case where K is a point, we obtain some new results about the topolog ical properties of f(R-q(f)), where R-q(f) is the set of points of M w here the rank of the differential of f is less than or equal to q. In particular, we show a result about the topological dimension of f(R-q( f)), which is closely related to a conjecture of Sard concerning the H ausdorff measure of f(R-q(f)).